FLC OF A B737 OVERSHOT ASSIGNED ALT DURING CLB DUE TO CONFUSION AND LACK OF CREW COORD OF SETTING ALT ALERTER. ARTCC RADAR CTLR INTERVENED AND REQUESTED THEY RETURN TO THE ACTUAL ASSIGNED ALT.

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ON CLBOUT FROM PHX WE WERE GIVEN 15000 FT AND HANDED OFF TO ZAB. ZAB STATED THAT FL280 WAS ROUGH AND ASKED IF WE WANTED TO LEVEL OFF LOWER; AT FL220. I THOUGHT I HEARD INSTRUCTIONS TO CLB TO FL280 AND THAT IT WAS ROUGH(TURB) BELOW FL280. I DIALED FL280 INTO THE ALT ALERTER AND CONTINUED THE CLB TO THAT ALT. AT APPROX FL235; THE ATC CTLR ASKED US TO VERIFY OUR ALT. WE STATED FL235 FOR FL280. THE CTLR STATED THAT OUR ASSIGNED ALT WAS FL220 AND TO EXPEDITE BACK TO FL220. I (PF) NEVER HEARD THE FL220 ASSIGNMENT; HOWEVER; THE CAPT DID RECOLLECT READING BACK FL220. I SHOULD HAVE VERIFIED THE ALT ASSIGNMENT WITH THE CAPT AFTER DIALING IN FL280 AND THIS CONFUSION WOULD NOT HAVE RESULTED. THE CTLR SAID NO CONFLICTS RESULTED AND VERIFIED THE CONFUSION ON ALT ASSIGNMENT STATING; 'I'M NOT SURE YOU READ BACK THE PROPER ALT; SO LET'S JUST FORGET IT.' SUPPLEMENTAL INFO FROM ACN 421637: ASKED BY ATC TWICE TO EXPEDITE CLB; 'THROUGH FL220 FOR FL280.' FREQ 124.85 ADVISED HE WOULD 'SHOW FL220 AS FINAL' BECAUSE OF TURB ABOVE. I'M NOT SURE IF THE CTLR CLRED US TO LEVEL AT FL220 AND I JUST DIDN'T GET IT INTO MY HEAD; OR WHETHER HE USED NONSTANDARD PHRASING AND DIDN'T GET HIS POINT ACROSS; BUT OUR ALT ALERTER REMAINED SET AT FL280; AND I AM USUALLY ADAMANT ABOUT ALT ALERTER SETTING PROCS AND XCHKS. I OFFERED TO PHONE CALL CTLR BUT HE DECLINED; SAID HE WASN'T SURE EXACTLY WHAT HAPPENED EITHER.